Are Infinite Points on a Straight Line Just One Point?

[JasonRox]

d(x,x)=0, what's the point of bringing this up? No one contested that.

You're saying it doesn't make sense to measure a point. That is a one right there.

 

[jing]

f(0) = 5
f(0+s) = 5
where s a very small fraction

No f(0+s) = 2s+5 and if s =/= 0 then f(0+s) =/= 5 no matter how small s is.

As s tends to 0 then f(0+s) tends to 5 but is only =5 when s=0

Look at JasonRox's example for 1/2^n n an integer

as n gets larger 1/2^n tends to 0 but no matter how large n is 1/2^n will still be none zero

 

[HallsofIvy]

Points have no length, so it is 0 length each point.

So, this doesn't make sense to you:

0+0+0+0+...?

The answer is clearly 0.

The sum of a finite number of 0s is 0. In fact, even the sum of a countable number of 0s is 0. But a line contains an uncountable the sum of an uncountable number of 0s is not 0- in fact, it's not defined.

 

[Werg22]

You're saying it doesn't make sense to measure a point. That is a one right there.

Let me clarify: it doesn't make sense to get the measure of a point for the purpose of using it as a part of a distance.

 

[JasonRox]

The sum of a finite number of 0s is 0. In fact, even the sum of a countable number of 0s is 0. But a line contains an uncountable the sum of an uncountable number of 0s is not 0- in fact, it's not defined.

That's what I talked about. I was just talking about that case.

 

[JasonRox]

This is where it gets tricky. I'm not 100% sure how to answer it, but logic fails here though.

Because what you're saying here is that you're counting all the zeroes and it adds to zero. The truth is that you can't even count all the zeroes! There are uncountably many points in a line, so you can't count them all even though they're all zero.

That's the post I'm talking about HallsofIvy.

The quote you posted has to do with werg22 talking about counting infinite many lengths not making sense to me, and so I commented...

Adding, 0+0+0+0+0+...=0, does not make sense?

That is infinitely many lengths in which in fact does make sense to add because there are countably many.